Let [ ] denote the greatest integer function. For a real number c , let g_c(x) = (x-c)[x^2] be defined for x (-2, 2) . Let f(c) denote the number of points in (-2, 2) where g_c(x) is discontinuous. If S is the set of points c (-2, 2) where the function f(c) is discontinuous, then the value of n(S) + _ c S c^2 is

Step-by-step solution

The function [x^2] is discontinuous at points where x^2 is an integer, provided x^2 crosses that integer value. For x (-2, 2) , x^2 [0, 4) . The integer values of x^2 are 0, 1, 2, 3 . At x = 0 , x^2 = 0 . For x close to 0 , x^2 > 0 but x^2 The points of discontinuity of [x^2] in (-2, 2) are where x^2 1, 2, 3 , which gives x 1, 2 , 3 . There are exactly 6 such points. The function g_c(x) = (x-c)[x^2] will be continuous at a point of discontinuity of [x^2] if and only if the factor (x-c) is zero at that point. Thus,